Variation in z

1 Intention

In this document, we are going to explore how diversity in species responses to one or both environmental variables, and different scenarios of correlation between them, may influence multifarious response diversity. Additionally, we are going to look how different correlation between the environmental variables influence the calculated response diversity.

Within the project “response diversity in the context of multifarious environmental change”, we have been simulating species response curves using the Eppley performance curve.

With one environmental variable, the performance (i.e., rate) is given by:

• \(rate(E) = ae^{bE}(1 - (\frac{E - z}{w/2})^2)\)
• \(E\) is the values of the environmental factor.
• \(z\) controls location of maximum.
• \(w\) controls range of \(E\) over which the rate is positive.
• \(a\) scaling constant.
• \(b\) controls rate of increase towards the maximum rate, as \(E\) increases.

Adding a second environmental variable gives:

\(rate(E_1, E_2) = a_1e^{b_1E_1}(1 - (\frac{E_1 - z_1}{w_1/2})^2) + a_2e^{b_2E_2}(1 - (\frac{E_2 - z_2}{w_2/2})^2)\)

We now want to assess the role of diversity in response to on (or both) environmental variable(s), and investigate how different covariances in diversity of responses to E1 and E2 influence response diversity.

To do that, we are going to manipulate amount of diversity in responses to either or both env variables among the species, and also the correlation in the tolerances (position of optima, determine by the terms z1 and z2 in the above formula). We will make scenarios of communities with diversity in response to only E1, or both E1 and E2, with diversity in both that is positively correlated (co-tolerance) or negatively (anti-tolerance) or no correlation.

2 Variation in diversity in z1

2.1 High fluctuation in environmental change - negative correlation

Increasing variance in z1 - negative correlation between env variables (E1 and E2) with high fluctuations We now create 3 communities. Community 1 is characterised by low diversity in z1, community 2 has medium diversity in z1, and community 3 has high diversity in z1. z2 is constant in the 3 communities and fixed to a medium level of diversity.

E1 and E2 have high fluctuations and negative correlation.

(ref:plot1env) Environmental variables with high variation and negative correlation. (a) E1 changing over time. (b) E2 changing over time.

Figure 2.1: (ref:plot1env)

2.1.1 Community 1 - low diversity in z1.

(ref:plotcomm1_neg) Community 1, single species responses. (a) Species responses to the gradient of E1. Different colour lines show the dependency of the rate to the second environmental variable (E2). (b) Species responses to the gradient of E2. Different colour lines show the dependency of the rate to the second environmental variable (E1).

(#fig:plotcomm1_neg)(ref:plotcomm1_neg)

2.1.2 Community 2 - medium diversity in z1.

(ref:plotcomm2_neg) Community 2, single species responses. (a) Species responses to the gradient of E1. Different colour lines show the dependency of the rate to the second environmental variable (E2). (b) Species responses to the gradient of E2. Different colour lines show the dependency of the rate to the second environmental variable (E1).

(#fig:plotcomm2_neg)(ref:plotcomm2_neg)

2.1.3 Community 3 - high diversity in z1.

(ref:plotcomm3_neg) Community 3, single species responses. (a) Species responses to the gradient of E1. Different colour lines show the dependency of the rate to the second environmental variable (E2). (b) Species responses to the gradient of E2. Different colour lines show the dependency of the rate to the second environmental variable (E1).

(#fig:plotcomm3_neg)(ref:plotcomm3_neg)

2.1.4 Response diversity calculation

(ref:RDplot1) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 2.2: (ref:RDplot1)

2.2 Low fluctuation in environmental change - negative correlation

Increasing variance in z1 - negative correlation between env variables (E1 and E2) with low fluctuations

We repeat the same steps as in the previous section, but this time E1 and E2 have a very small variance.

##              E1         E2       time
## E1    1.0000000 -1.0000000 -0.3694144
## E2   -1.0000000  1.0000000  0.3694144
## time -0.3694144  0.3694144  1.0000000

Figure 2.3: Environmental variables with low variation and negative correlation. (a) E1 changing over time. (b) E2 changing over time.

2.2.1 Response diversity calculation

(ref:RDplot2) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 2.4: (ref:RDplot1)

2.3 High fluctuation in environmental change - positive correlation

Increasing variance in z1 - positive correlation between env variables (E1 and E2) with high fluctuations

Same steps has before (3 communities with increasing diversity ib z1 while z2 is fixed), but the correlation between E1 and E1 is negative.

(ref:plot3env) Environmental variables with high variation and positive correlation. (a) E1 changing over time. (b) E2 changing over time.

Figure 2.5: (ref:plot3env)

2.3.1 Response diversity calculation

(ref:RDplot3) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 2.6: (ref:RDplot3)

2.4 Low fluctuation in environmental change - positive correlation

Increasing variance in z1 - positive correlation between env variables (E1 and E2) with low fluctuations

We repeat the same steps as in the previous section, but this time E1 and E2 have a very small variance.

Environmental variables with low variation and negative correlation. (a) E1 changing over time. (b) E2 changing over time. Environmental variables with low variation and negative correlation. (a) E1 changing over time. (b) E2 changing over time.

##             E1        E2      time
## E1   1.0000000 1.0000000 0.2678652
## E2   1.0000000 1.0000000 0.2678652
## time 0.2678652 0.2678652 1.0000000

Figure 2.7: (ref:plot4env)

2.4.1 Response diversity calculation

(ref:RDplot4) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 2.8: (ref:RDplot4)

2.5 High fluctuation in environmental change - no correlation

Increasing variance in z1 - no correlation between env variables (E1 and E2) with high fluctuations

Same steps has before (3 communities with increasing diversity ib z1 while z2 is fixed), but there is no correlation between E1 and E1

(ref:plot5env) Environmental variables with high variation and no correlation. (a) E1 changing over time. (b) E2 changing over time.

Figure 2.9: (ref:plot5env)

2.5.1 Response diversity calculation

(ref:RDplot5) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 2.10: (ref:RDplot5)

2.6 Low fluctuation in environmental change - no correlation

Increasing variance in z1 - no correlation between env variables (E1 and E2) with low fluctuations

We repeat the same steps as in the previous section, but this time E1 and E2 have a very small variance.

##               E1          E2       time
## E1    1.00000000 -0.06610452 0.03353153
## E2   -0.06610452  1.00000000 0.26786523
## time  0.03353153  0.26786523 1.00000000

Figure 2.11: Environmental variables with low variation and negative correlation. (a) E1 changing over time. (b) E2 changing over time.

##   sp time   E1_ref   E2_ref     dir_deriv
## 1 s1    1 297.4660 26.24653 -3.672372e-09
## 2 s1    2 298.6443 23.47597 -2.026398e-09
## 3 s1    3 296.9343 24.10300  2.124898e-09
## 4 s1    4 300.6952 24.29208 -1.826044e-09
## 5 s1    5 297.9256 24.85162  2.487754e-09
## 6 s1    6 298.0572 26.49562 -5.399204e-09

##   sp time   E1_ref   E2_ref     dir_deriv
## 1 s1    1 297.4660 26.24653 -3.149255e-09
## 2 s1    2 298.6443 23.47597 -1.675578e-09
## 3 s1    3 296.9343 24.10300  1.787855e-09
## 4 s1    4 300.6952 24.29208 -1.513506e-09
## 5 s1    5 297.9256 24.85162  2.161479e-09
## 6 s1    6 298.0572 26.49562 -4.610565e-09

2.6.1 Response diversity calculation

(ref:RDplot6) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 2.12: (ref:RDplot6)

3 Variation in diversity in z1 and z2

Now we do the same, but having also z2 changing together with z1 with positive correlation. So, diversity in z1 and z2 are increasing gradually from low to high in the 3 communities. I expect RD to increase when both z1 and z2 are high and there is negative correlation in env change (E1 increases when E2 decreases).

3.1 High fluctuation in environmental change - negative correlation

Increasing variance in z1 and z2 - negative correlation between env variables (E1 and E2) with high fluctuations We now create 3 communities. Community 1 is characterised by low diversity in z1 and z2, community 2 has medium diversity in z1 and z2, and community 3 has high diversity in z1 and z2.

E1 and E2 have high fluctuations and negative correlation.

(ref:plot7env) Environmental variables with high variation and negative correlation. (a) E1 changing over time. (b) E2 changing over time.

Figure 3.1: (ref:plot7env)

3.1.1 Community 1 - low diversity in z1 and z2.

(ref:plotcomm1.2_neg) Community 1, single species responses. (a) Species responses to the gradient of E1. Different colour lines show the dependency of the rate to the second environmental variable (E2). (b) Species responses to the gradient of E2. Different colour lines show the dependency of the rate to the second environmental variable (E1).

(#fig:plotcomm1.2_neg)(ref:plotcomm1.2_neg)

3.1.2 Community 2 - medium diversity in z1 and z2.

(ref:plotcomm2.1_neg) Community 2, single species responses. (a) Species responses to the gradient of E1. Different colour lines show the dependency of the rate to the second environmental variable (E2). (b) Species responses to the gradient of E2. Different colour lines show the dependency of the rate to the second environmental variable (E1).

(#fig:plotcomm2.1_neg)(ref:plotcomm2.1_neg)

3.1.3 Community 3 - high diversity in z1 and z2.

(ref:plotcomm3.2_neg) Community 3, single species responses. (a) Species responses to the gradient of E1. Different colour lines show the dependency of the rate to the second environmental variable (E2). (b) Species responses to the gradient of E2. Different colour lines show the dependency of the rate to the second environmental variable (E1).

(#fig:plotcomm3.2_neg)(ref:plotcomm3.2_neg)

3.1.4 Response diversity calculation

(ref:RDplot7) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 3.2: (ref:RDplot7)

3.2 Low fluctuation in environmental change - negative correlation

Increasing variance in z1 and z2 - negative correlation between env variables (E1 and E2) with low fluctuations

We repeat the same steps as in the previous section, but this time E1 and E2 have a very small variance.

##              E1         E2       time
## E1    1.0000000 -1.0000000 -0.3694144
## E2   -1.0000000  1.0000000  0.3694144
## time -0.3694144  0.3694144  1.0000000

Figure 3.3: Environmental variables with low variation and negative correlation. (a) E1 changing over time. (b) E2 changing over time.

3.2.1 Response diversity calculation

(ref:RDplot8) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 3.4: (ref:RDplot8)

3.3 High fluctuation in environmental change - positive correlation

Increasing variance in z1 and z2 - positive correlation between env variables (E1 and E2) with high fluctuations

Same steps has before (3 communities with increasing diversity in z1 and z2), but there is positive correlation between E1 and E1

##             E1        E2      time
## E1   1.0000000 1.0000000 0.2678652
## E2   1.0000000 1.0000000 0.2678652
## time 0.2678652 0.2678652 1.0000000

Figure 3.5: Environmental variables with high variation and positive correlation. (a) E1 changing over time. (b) E2 changing over time.

3.3.1 Response diversity calculation

(ref:RDplot9) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 3.6: (ref:RDplot9)

3.4 Low fluctuation in environmental change - positive correlation

Increasing variance in z1 and z2 - positive correlation between env variables (E1 and E2) with low fluctuations

We repeat the same steps as in the previous section, but this time E1 and E2 have a very small variance.

(ref:plot10env) Environmental variables with low variation and positive correlation. (a) E1 changing over time. (b) E2 changing over time.

Figure 3.7: (ref:plot10env)

##   sp time   E1_ref   E2_ref     dir_deriv
## 1 s1    1 299.3965 26.24653 -5.098769e-09
## 2 s1    2 296.6260 23.47597  1.131052e-09
## 3 s1    3 297.2530 24.10300  1.948664e-09
## 4 s1    4 297.4421 24.29208  2.211908e-09
## 5 s1    5 298.0016 24.85162  3.036941e-09
## 6 s1    6 299.6456 26.49562 -5.432922e-09

3.4.1 Response diversity calculation

(ref:RDplot10) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 3.8: (ref:RDplot10)

3.5 High fluctuation in environmental change - no correlation

Increasing variance in z1 and z2 - no correlation between env variables (E1 and E2) with high fluctuations

Same steps has before (3 communities with increasing diversity in z1 and z2), but there is no correlation between E1 and E1

##               E1         E2        time
## E1    1.00000000 0.06610452 -0.26786523
## E2    0.06610452 1.00000000  0.03353153
## time -0.26786523 0.03353153  1.00000000

Figure 3.9: Environmental variables with high variation and no correlation. (a) E1 changing over time. (b) E2 changing over time.

##   sp time   E1_ref   E2_ref     dir_deriv
## 1 s1    1 282.8832 18.51107  3.963815e-10
## 2 s1    2 316.8155 29.68894 -1.135476e-08
## 3 s1    3 309.1360 13.46697 -4.402670e-09
## 4 s1    4 306.8202 49.14590  1.013249e-08
## 5 s1    5 299.9673 22.87156 -2.005458e-09
## 6 s1    6 279.8325 24.11935  1.797159e-09

##   sp time   E1_ref   E2_ref     dir_deriv
## 1 s1    1 282.8832 18.51107  4.996893e-10
## 2 s1    2 316.8155 29.68894 -1.055866e-08
## 3 s1    3 309.1360 13.46697 -4.022972e-09
## 4 s1    4 306.8202 49.14590  9.482413e-09
## 5 s1    5 299.9673 22.87156 -1.813781e-09
## 6 s1    6 279.8325 24.11935  1.650946e-09

3.5.1 Response diversity calculation

(ref:RDplot11) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 3.10: (ref:RDplot11)

3.6 Low fluctuation in environmental change - no correlation

Increasing variance in z1 and z2 - no correlation between env variables (E1 and E2) with low fluctuations

We repeat the same steps as in the previous section, but this time E1 and E2 have a very small variance.

(ref:plot12env) Environmental variables with low variation and negative correlation. (a) E1 changing over time. (b) E2 changing over time.

Figure 3.11: (ref:plot12env)

##   sp time   E1_ref   E2_ref     dir_deriv
## 1 s1    1 297.4660 26.24653 -3.277762e-09
## 2 s1    2 298.6443 23.47597 -1.722922e-09
## 3 s1    3 296.9343 24.10300  1.852674e-09
## 4 s1    4 300.6952 24.29208 -1.559120e-09
## 5 s1    5 297.9256 24.85162  2.265060e-09
## 6 s1    6 298.0572 26.49562 -4.790500e-09

3.6.1 Response diversity calculation

(ref:RDplot12) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 3.12: (ref:RDplot12)

4 Variation in diversity in z1 and z2

Now we do the same, but having also z2 changing together with z1 with negative correlation. So, diversity in z1 is increasing gradually from low to high in the 3 communities, while z2 decreases gradually in the 3 communities.

I expect RD to be the highest when either z1 or z2 is high and there is negative correlation in env change (E1 increases when E2 decreases).

4.1 High fluctuation in environmental change - negative correlation

Increasing variance in z1 and z2 - negative correlation between env variables (E1 and E2) with high fluctuations We now create 3 communities. Community 1 is characterised by low diversity in z1 and high diversity in z2, community 2 has medium diversity in z1 and z2, and community 3 has high diversity in z1 and low in z2.

E1 and E2 have high fluctuations and negative correlation.

(ref:plot13env) Environmental variables with high variation and negative correlation. (a) E1 changing over time. (b) E2 changing over time.

Figure 4.1: (ref:plot13env)

4.1.1 Community 1 - low diversity in z1 and high diversity in z2.

(ref:plotcomm1.3_neg) Community 1, single species responses. (a) Species responses to the gradient of E1. Different colour lines show the dependency of the rate to the second environmental variable (E2). (b) Species responses to the gradient of E2. Different colour lines show the dependency of the rate to the second environmental variable (E1).

(#fig:plotcomm1.3_neg)(ref:plotcomm1.3_neg)

4.1.2 Community 2 - medium diversity in z1 and z2.

(ref:plotcomm2.3_neg) Community 2, single species responses. (a) Species responses to the gradient of E1. Different colour lines show the dependency of the rate to the second environmental variable (E2). (b) Species responses to the gradient of E2. Different colour lines show the dependency of the rate to the second environmental variable (E1).

(#fig:plotcomm2.3_neg)(ref:plotcomm2.3_neg)

4.1.3 Community 3 - high diversity in z1 and low in z2.

(ref:plotcomm3.3_neg) Community 3, single species responses. (a) Species responses to the gradient of E1. Different colour lines show the dependency of the rate to the second environmental variable (E2). (b) Species responses to the gradient of E2. Different colour lines show the dependency of the rate to the second environmental variable (E1).

(#fig:plotcomm3.3_neg)(ref:plotcomm3.3_neg)

4.1.4 Response diversity calculation

(ref:RDplot13) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 4.2: (ref:RDplot13)

4.2 Low fluctuation in environmental change - negative correlation

Increasing variance in z1 and z2 - negative correlation between env variables (E1 and E2) with low fluctuations

We repeat the same steps as in the previous section, but this time E1 and E2 have a very small variance.

(ref:plot14env) Environmental variables with low variation and negative correlation. (a) E1 changing over time. (b) E2 changing over time.

Figure 4.3: (ref:plot14env)

4.2.1 Response diversity calculation

(ref:RDplot14) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 4.4: (ref:RDplot14)

4.3 High fluctuation in environmental change - positive correlation

Increasing variance in z1 and decreasing in z2 - positive correlation between env variables (E1 and E2) with high fluctuations

Same steps has before (3 communities with increasing diversity in z1 and z2), but there is positive correlation between E1 and E1

(ref:plot15env) Environmental variables with high variation and positive correlation. (a) E1 changing over time. (b) E2 changing over time.

Figure 4.5: (ref:plot15env)

4.3.1 Response diversity calculation

(ref:RDplot15) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 4.6: (ref:RDplot15)

4.4 Low fluctuation in environmental change - positive correlation

Increasing variance in z1 and decreasing in 2 - positive correlation between env variables (E1 and E2) with low fluctuations.

We repeat the same steps as in the previous section, but this time E1 and E2 have a very small variance.

(ref:plot16env) Environmental variables with low variation and positive correlation. (a) E1 changing over time. (b) E2 changing over time.

Figure 4.7: (ref:plot16env)

4.4.1 Response diversity calculation

(ref:RDplot16) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 4.8: (ref:RDplot16)

4.5 High fluctuation in environmental change - no correlation

Increasing variance in z1 and decreasing in z2 - no correlation between env variables (E1 and E2) with high fluctuations

Same steps has before (3 communities with increasing diversity in z1 and z2), but there is no correlation between E1 and E1

(ref:plot17env) Environmental variables with high variation and no correlation. (a) E1 changing over time. (b) E2 changing over time.

Figure 4.9: (ref:plot17env)

4.5.1 Response diversity calculation

(ref:RDplot17) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 4.10: (ref:RDplot17)

4.6 Low fluctuation in environmental change - no correlation

Increasing variance in z1 and decreasing in z2 - no correlation between env variables (E1 and E2) with low fluctuations

We repeat the same steps as in the previous section, but this time E1 and E2 have a very small variance.

(ref:plot18env) Environmental variables with low variation and negative correlation. (a) E1 changing over time. (b) E2 changing over time.

Figure 4.11: (ref:plot18env)

4.6.1 Response diversity calculation

(ref:RDplot18) Directional derivatives and response diversity for the three different communities. a, b, c. Species directional derivatives over time. d, e, f. Response diversity measured as similarity-based diversity metric. g, h, i. Response diversity measured as divergence (sign sensitive).

Figure 4.12: (ref:RDplot18)